Hamiltonian dynamics of vortex and magnetic lines in hydrodynamic type systems

Vortex line and magnetic line representations are introduced for a description of flows in ideal hydrodynamics and magnetohydrodynamics (MHD), respectively. For incompressible fluids, it is shown with the help of this transformation that the equations of motion for vorticity $\mathit{\Omega }$ and magnetic field follow from a variational principle. By means of this representation, it is possible to integrate the hydrodynamic type system with the Hamiltonian $\mathcal{H}=\int |\mathit{\Omega }|d\mathbf{r}$ and some other systems. It is also demonstrated that these representations allow one to remove from the noncanonical Poisson brackets, defined in the space of divergence-free vector fields, the degeneracy connected with the vorticity frozenness for the Euler equation and with magnetic field frozenness for ideal MHD. For MHD, a new Weber-type transformation is found. It is shown how this transformation can be obtained from the two-fluid model when electrons and ions can be considered as two independent fluids. The Weber-type transformation for ideal MHD gives the whole Lagrangian vector invariant. When this invariant is absent, this transformation coincides with the Clebsch representation analog introduced by V.E. Zakharov and E. A. Kuznetsov [Dokl. Ajad. Nauk 194, 1288 (1970) [Sov. Phys. Dokl. 15, 913 (1971)]].